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The boundaries of quantum materials can host a variety of exotic effects such as topologically robust edge states or anyonic quasiparticles. Here, we show that fermionic systems such as graphene that admit a low energy Dirac description can exhibit counterintuitive relativistic effects at their boundaries. As an example, we consider carbon nanotubes and demonstrate that relativistic bulk spinor states can have non zero charge density on the boundaries, in contrast to the sinusoidal distribution of non-relativistic wave functions that are necessarily zero at the boundaries. This unusual property of relativistic spinors is complementary to the linear energy dispersion relation exhibited by Dirac materials and can influence their coupling to leads, transport properties or their response to external fields., Comment: 9 pages, 8 figures

Systems of interacting fermions can give rise to ground states whose correlations become effectively free-fermion-like in the thermodynamic limit, as shown by Baxter for a class of integrable models that include the one-dimensional XYZ spin-$\frac{1}{2}$ chain. Here, we quantitatively analyse this behaviour by establishing the relation between system size and correlation length required for the fermionic Gaussianity to emerge. Importantly, we demonstrate that this behaviour can be observed through the applicability of Wick's theorem and thus it is experimentally accessible. To establish the relevance of our results to possible experimental realisations of XYZ-related models, we demonstrate that the emergent Gaussianity is insensitive to weak variations in the range of interactions, coupling inhomogeneities and local random potentials., Comment: 8 pages, 4 figures

Here, we analyse two Dirac fermion species in two spatial dimensions in the presence of general quartic contact interactions. By employing functional bosonisation techniques, we demonstrate that depending on the couplings of the fermion interactions the system can be effectively described by a rich variety of topologically massive gauge theories. Among these effective theories, we obtain an extended Chern–Simons theory with higher order derivatives as well as two coupled Chern–Simons theories. Our formalism allows for a general description of interacting fermions emerging, for example, at the gapped boundary of three-dimensional topological crystalline insulators., SCOPUS: ar.j, info:eu-repo/semantics/published

Quasiparticle poisoning, expected to arise during the measurement of Majorana zero mode state, poses a fundamental problem towards the realization of Majorana-based quantum computation. Parafermions, a natural generalization of Majorana fermions, can encode topological qudits immune to quasiparticle poisoning. While parafermions are expected to emerge in superconducting fractional quantum Hall systems, they are not yet attainable with current technology. To bypass this problem, we employ a photonic quantum simulator to experimentally demonstrate the key components of parafermion-based universal quantum computation. Our contributions in this article are twofold. First, by manipulating the photonic states, we realize Clifford operator Berry phases that correspond to braiding statistics of parafermions. Second, we investigate the quantum contextuality in a topological system for the first time by demonstrating the contextuality of parafermion encoded qudit states. Importantly, we find that the topologically-encoded contextuality opens the way to magic state distillation, while both the contextuality and the braiding-induced Clifford gates are resilient against local noise. By introducing contextuality, our photonic quantum simulation provides the first step towards a physically robust methodology for realizing topological quantum computation., 11+6 pages, 5+2 figures, 1+2 tables, presentation extended and improved, analysis and results the same, to appear in PRX Quantum

We demonstrate that $\mathbb{Z}_2$ gauge transformations and lattice deformations in Kitaev's honeycomb lattice model can have the same description in the continuum limit of the model in terms of chiral gauge fields. The chiral gauge fields are coupled to the Majorana fermions that satisfy the Dirac dispersion relation in the non-Abelian sector of the model. For particular values, the effective chiral gauge field becomes equivalent to the $\mathbb{Z}_2$ gauge field, enabling us to associate effective fluxes to lattice deformations. Motivated by this equivalence, we consider Majorana-bounding $\pi$ vortices and Majorana-bounding lattice twists and demonstrate that they are adiabatically connected to each other. This equivalence opens the possibility for novel encoding of Majorana-bounding defects that might be easier to realise in experiments., Comment: 17 pages, 11 figures

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents. Nevertheless, it is not known how accurately curved geometry can describe actual microscopic models. Here, we demonstrate that the low-energy properties of the Kitaev honeycomb lattice model, a topological superconductor that supports localised Majorana zero modes at its vortex excitations, are faithfully described in terms of Riemann-Cartan geometry. In particular, we show analytically that the continuum limit of the model is given in terms of the Majorana version of the Dirac Hamiltonian coupled to both curvature and torsion. We numerically establish the accuracy of the geometric description for a wide variety of couplings of the microscopic model. Our work opens up the opportunity to accurately predict dynamical properties of the Kitaev model from its effective geometric description., 16 pages, 10 figures

5 pags., 5 figs., Here we provide a general methodology to directly measure the topological currents emerging in the optical lattice implementation of the Haldane model. Alongside the edge currents supported by gapless edge states, transverse currents can emerge in the bulk of the system whenever the local potential is varied in space, even if it does not cause a phase transition. In optical lattice implementations the overall harmonic potential that traps the atoms provides the boundaries of the topological phase that supports the edge currents, as well as providing the potential gradient across the topological phase that gives rise to the bulk current. Both the edge and bulk currents are resilient to several experimental parameters such as trapping potential, temperature, and disorder. We propose to investigate the properties of these currents directly from time-of-flight images with both short-time and long-time expansions., This work was supported by the EPSRC Grant No. EP/R020612/1; Spanish Projects PGC2018-094792-B-I00 (MCIU/AEI/FEDER, EU), PGC2018-094180-B-I00 (MCIU /AEI/FEDER, EU), and FIS2015-63770-P (MINECO /FEDER, EU); CAM/FEDER Project No. S2018/TCS-4342 (QUITEMAD-CM), and CSIC Research Platform PTI-001.

We employ the interaction distance to characterise the physics of a one-dimensional extended XXZ spin model, whose phase diagram consists of both integrable and non-integrable regimes, with various types of ordering, e.g., a gapless Luttinger liquid and gapped crystalline phases. We numerically demonstrate that the interaction distance successfully reveals the known behaviour of the model in its integrable regime. As an additional diagnostic tool, we introduce the notion of "integrability distance" and particularise it to the XXZ model in order to quantity how far the ground state of the extended XXZ model is from being integrable. This distance provides insight into the properties of the gapless Luttinger liquid phase in the presence of next-nearest neighbour spin interactions which break integrability., 12 pages, 9 figures

Majorana fermions are a fascinating medium for discovering new phases of matter. However, the standard analytical tools are very limited in probing the non-perturbative aspects of interacting Majoranas in more than one dimensions. Here, we employ the holographic correspondence to determine the specific heat of a two-dimensional interacting gapless Majorana system. To perform our analysis we first describe the interactions in terms of a pseudo-scalar torsion field. We then allow fluctuations in the background curvature thus identifying our model with a (2 + 1)-dimensional Anti-de Sitter (AdS) geometry with torsion. By employing the AdS/CFT correspondence, we show that the interacting model is dual to a (1 + 1)-dimensional conformal field theory (CFT) with central charge that depends on the interaction coupling. This non-perturbative result enables us to determine the effect interactions have in the specific heat of the system at the zero temperature limit., SCOPUS: ar.j, info:eu-repo/semantics/published

10 pags., 8 figs., 1 tab., 1 app., We provide evidence that, alongside topologically protected edge states, two-dimensional Chern insulators also support localized bulk states deep in their valence and conduction bands. These states manifest when local potential gradients are applied to the bulk, while all parts of the system remain adiabatically connected to the same phase. In turn, the bulk states produce bulk current transverse to the potential difference. This occurs even when the potential is always below the energy gap, where one expects only edge currents to appear. Bulk currents are topologically protected and behave as edge currents under an external influence, such as temperature or local disorder. Detecting topologically resilient bulk currents offers a direct means to probe the localized bulk states., This work was supported by the EPSRC Grant No. EP/R020612/1, Spanish Projects No. PGC2018-094792-B-I00 (MCIU/AEI/FEDER, EU), No. PGC2018-094180-B-I00 (MCIU/AEI/FEDER, EU), No. FIS2015-63770-P (MINECO/FEDER, EU), CAM/FEDER Project No. S2018/TCS-4342 (QUITEMAD-CM), and CSIC Research Platform PTI-001.

Free fermion systems enjoy a privileged place in physics. With their simple structure they can explain a variety of effects, ranging from insulating and metallic behaviours to superconductivity and the integer quantum Hall effect. Interactions, e.g. in the form of Coulomb repulsion, can dramatically alter this picture by giving rise to emerging physics that may not resemble free fermions. Examples of such phenomena include high-temperature superconductivity, fractional quantum Hall effect, Kondo effect and quantum spin liquids. The non-perturbative behaviour of such systems remains a major obstacle to their theoretical understanding that could unlock further technological applications. Here, we present a pedagogical review of "interaction distance" [Nat. Commun. 8, 14926 (2017)] -- a systematic method that quantifies the effect interactions can have on the energy spectrum and on the quantum correlations of generic many-body systems. In particular, the interaction distance is a diagnostic tool that identifies the emergent physics of interacting systems. We illustrate this method on the simple example of a one-dimensional Fermi-Hubbard dimer., Comment: 27 pages, 8 figures; submission to SciPost

Thermalisation is a probabilistic process. As such, it is generally expected that when we increase the temperature of a system its classical behaviour dominates its quantum coherences. By employing the Gibbs state of a translationally invariant quantum spin liquid - Kitaev's honeycomb lattice model - we demonstrate that an insulating phase at $T=0$ becomes metallic purely by increasing temperature. In particular, we compute the finite temperature distribution of energies and show that it diverges logarithmically, as we move to small energies. The corresponding wave functions become critical alike as at Anderson transitions. These characteristics are obtained within a new exact Monte Carlo method that simulates the finite temperature behaviour of the Kitaev model. In particular, we take into account the projection onto the physical parity sectors, required for identifying the topological degeneracy of the model. Our work opens the possibility to detect thermal metal behaviour in spin liquid experiments., Comment: some content rearranged, small text changes

Geometric phases, generated by cyclic evolutions of quantum systems, offer an inspiring playground for advancing fundamental physics and technologies, alike. Intriguingly, the exotic statistics of anyons realised in physical systems can be interpreted as a topological version of geometric phases. However, non-Abelian statistics has not yet been demonstrated in the laboratory. Here we employ an all optical quantum system that simulates the statistical evolution of Majorana fermions. As a result we experimentally realise non-Abelian Berry phases with the topological characteristic that they are invariant under continues deformations of their control parameters. We implement a universal set of Majorana inspired gates by performing topological and non-topological evolutions and investigate their resilience against perturbative errors. Our photonic experiment, while it is not scalable, it suggests the intriguing possibility of experimentally simulating Majorana statistics with scalable technologies., Comment: 8 pages, 4 figures

Density functional theory maps an interacting Hamiltonian onto the Kohn-Sham Hamiltonian, an explicitly free model with identical local fermion densities. Using the interaction distance, the minimum distance between the ground state of the interacting system and a generic free fermion state, we quantify the applicability and limitations of the exact Kohn-Sham model in capturing the various properties of the interacting system. As a byproduct, this distance determines the optimal free state that reproduces the entanglement properties of the interacting system as faithfully as possible. The parent Hamiltonian of the optimal free state identifies a system that can determine the expectation value of any observable with controlled accuracy. This optimal entanglement model opens up the possibility of extending the systematic applicability of auxiliary free models into the non-perturbative, strongly-correlated regimes., Comment: 7+2 pages, 3+1 figures, presentation improved

The boundary of a 2D topological superconductor can be modeled by a conformal field theory. Here we demonstrate the behaviors of this high level description emerging from a microscopic model at finite temperatures. To achieve that, we analyze the low energy sector of Kitaev's honeycomb lattice model and probe its energy current. We observe that the scaling of the energy current with temperature reveals the central charge of the conformal field theory, which is in agreement with the Chern number of the bulk. Importantly, these currents can discriminate between distinct topological phases at finite temperatures. We assess the resilience of this measurement of the central charge under coupling disorder, bulk dimerisation and defects at the boundary, thus establishing it as a favorable means of experimentally probing topological superconductors., Comment: updated -- minor tweaks to text and figures

We experimentally simulate the exchange of Majorana zero modes with a photonic quantum simulator. Our work opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.

Helical Majorana edge states at the 2D boundaries of 3D topological superconductors can be gapped by a surface Zeeman field. Here we study the effect nested defects imprinted on the Zeeman field can have on the edge states. We demonstrate that depending on the configuration of the field we can induce dimensional reduction of gapless Majorana modes from 2D to 1D or quasi-0D at magnetic domain walls. We determine the nature of the Majorana localisation on these defects as a function of the magnitude and configuration of the Zeeman field. Finally, we observe a generalisation of the index theorem governing the number of gapless modes at the interface between topologically non-trivial systems with partial Chern numbers., 11 pages, 9 figures

We evaluate the entanglement entropy of exactly solvable Hamiltonians corresponding to general families of three-dimensional topological models. We show that the modification to the entropic area law due to three-dimensional topological properties is richer than the two-dimensional case. In addition to the reduction of the entropy caused by a nonzero vacuum expectation value of contractible loop operators, a topological invariant emerges that increases the entropy if the model consists of nontrivially braiding anyons. As a result the three-dimensional topological entanglement entropy provides only partial information about the two entropic topological invariants.

We consider topological quantum memories for a general class of abelian anyon models defined on spin lattices. These are non-universal for quantum computation when restricting to topological operations alone, such as braiding and fusion. The effects of additional non-topological operations, such as spin measurements, are studied. These are shown to allow universal quantum computation, while still utilizing topological protection. Our work gives an insight into the relation between abelian models and their non-abelian counterparts., 10 pages, 3 figures, presented at 6th QPL workshop

A study of the thermal properties of two-dimensional topological lattice models is presented. This work is relevant to assess the usefulness of these systems as a quantum memory. For our purposes, we use the topological mutual information $I_{\mathrm{topo}}$ as a "topological order parameter". For Abelian models, we show how $I_{\mathrm{topo}}$ depends on the thermal topological charge probability distribution. More generally, we present a conjecture that $I_{\mathrm{topo}}$ can (asymptotically) be written as a Kullback-Leitner distance between this probability distribution and that induced by the quantum dimensions of the model at hand. We also explain why $I_{\mathrm{topo}}$ is more suitable for our purposes than the more familiar entanglement entropy $S_{\mathrm{topo}}$. A scaling law, encoding the interplay of volume and temperature effects, as well as different limit procedures, are derived in detail. A non-Abelian model is next analysed and similar results are found. Finally, we also consider, in the case of a one-plaquette toric code, an environment model giving rise to a simulation of thermal effects in time.

A doublet of three-dimensional Dirac fermions can effectively describe the low energy spectrum of a fermionic cubic lattice. We employ this fermion doubling to encode a non-Abelian SU(2) charge in the fundamental representation. We explicitly demonstrate that suitable distortion of the tunnelling couplings can introduce a scalar and a Yang-Mills field in the effective low energy description, both coupled to the Dirac fermions. The simplicity of the model suggests its physical implementation with ultra-cold atoms or molecules.

Fox’s H-function provide a unified and elegant framework to tackle several physical phenomena. We solve the space fractional diffusion equation on the real line equipped with a delta distribution initial condition and identify the corresponding H-function by studying the small x expansion of the solution. The asymptotic expansions near zero and infinity are expressed, for rational values of the index α , in terms of a finite series of generalized hypergeometric functions. In x-space, the α = 1 stable law is also derived by solving the anomalous diffusion equation with an appropriately chosen infinitesimal generator for time translations. We propose a new classification scheme of stable laws according to which a stable law is now characterized by a generating probability density function. Knowing this elementary probability density function and bearing in mind the infinitely divisible property we can reconstruct the corresponding stable law. Finally, using the asymptotic behavior of H-function in terms of hypergeometric functions we can compute closed expressions for the probability density functions depending on their parameters α , β , c , τ . Known cases are then reproduced and new probability density functions are presented.

Topological phases of matter possess intricate correlation patterns typically probed by entanglement entropies or entanglement spectra. In this work, we propose an alternative approach to assessing topologically induced edge states in free and interacting fermionic systems. We do so by focussing on the fermionic covariance matrix. This matrix is often tractable either analytically or numerically and it precisely captures the relevant correlations of the system. By invoking the concept of monogamy of entanglement we show that highly entangled states supported across a system bi-partition are largely disentangled from the rest of the system, thus appearing usually as gapless edge states. We then define an entanglement qualifier that identifies the presence of topological edge states based purely on correlations present in the ground states. We demonstrate the versatility of this qualifier by applying it to various free and interacting fermionic topological systems., 4+3 pages, 3+3 figures

Summarization: This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers. Presented on

We analytically derive a compatible family of effective field theories that uniquely describe topological superconductors in 3D, their 2D boundary and their 1D defect lines. We start by deriving the topological field theory of a 3D topological superconductor in class DIII, which is consistent with its symmetries. Then we identify the effective theory of a 2D topological superconductor in class D living on the gapped boundary of the 3D system. By employing the holographic correspondence we derive the effective chiral conformal field theory that describes the gapless modes living on the defect lines or effective boundary of the class D topological superconductor. We demonstrate that the chiral central charge is given in terms of the 3D winding number of the bulk which by its turn is equal to the Chern number of its gapped boundary., 4 pages

The entanglement properties of some novel quantum systems are studied that are inspired by recent developments in cold-atom technology. A triangular optical lattice of two atomic species can be employed to generate a variety of spin-1/2 Hamiltonians including effective three-spin interactions. A variety of one or two dimensional systems can thus be realized that possess multi-degenerate ground states or non-vanishing chirality. The properties of these ground states and their phase transitions are probed with appropriate measures such as the entropic entanglement and the spin chirality., 9 pages, 6 figures, talk presented at the De Giorgi Workshop on Entanglement in Physical and Information Science, Pisa 2004

We present an idealized model involving interacting quantum dots that can support both the dynamical and geometrical forms of quantum computation. We show that by employing a structure similar to the one used in the Aharonov-Bohm effect we can construct a topological two-qubit phase-gate that is to a large degree independent of the exact values of the control parameters and therefore resilient to control errors. The main components of the setup are realizable with present technology., 8 pages, 3 figures, submitted to Jour. of Opt. B (special issue on Quantum Computing)

We discuss an experimental setup where two laser-driven atoms spontaneously emit photons and every photon causes a ``click'' at a point on a screen. By deriving the probability density for an emission into a certain direction from basic quantum mechanical principles we predict a spatial interference pattern. Similarities and differences with the classical double-slit experiment are discussed., Comment: 6 pages, 2 figures, contribution to the symposium "100 Years of Werner Heisenberg - Works and Impact" in Bamberg, to appear in Fortschritte der Physik

The realization of Majorana zero modes is in the centre of intense theoretical and experimental investigations. Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan-Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin-1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a three-site chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures., main text (10 pages, 4 figures) and detailed supplementary information

The quasi one-dimensional transport of Abelian and non-Abelian anyons is studied in the presence of a random topological background. In particular, we consider the quantum walk of an anyon that braids around islands of randomly filled static anyons of the same type. Two distinct behaviours are identified. We analytically demonstrate that all types of Abelian anyons localise purely due to the statistical phases induced by their random anyonic environment. In contrast, we numerically show that non-Abelian Ising anyons do not localise. This is due to their entanglement with the anyonic environment that effectively induces dephasing. Our study demonstrates that localisation properties strongly depend on non-local topological interactions and it provides a clear distinction in the transport properties of Abelian and non-Abelian statistics., Comment: 9 pages, 5 figures

Currently, there is much interest in discovering analytically tractable $(3+1)$-dimensional models that describe interacting fermions with emerging topological properties. Towards that end we present a three-dimensional tight-binding model of spinless interacting fermions that reproduces, in the low-energy limit, a $(3+1)$-dimensional Abelian topological quantum field theory called the BF model. By employing a mechanism equivalent to Haldane's Chern insulator, we can turn the noninteracting model into a three-dimensional chiral topological insulator. We then isolate energetically one of the two Fermi points of the lattice model. In the presence of suitable fermionic interactions, the system, in the continuum limit, is equivalent to a generalized $(3+1)$-dimensional Thirring model. The low-energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of $(2+1)$-dimensional BF theory at the boundary of the lattice and it provides a way to detect the topological order of the model through fermionic density measurements.

20 pags.; 3 figs.; app. Creative Commons Attribution 3.0 licence, Topological invariants, such as the Chern number, characterize topological phases of matter. Here we provide a method to detect Chern numbers in systems with two distinct species of fermion, such as spins, orbitals or several atomic states. We analytically show that the Chern number can be decomposed as a sum of component specific winding numbers, which are themselves physically observable. We apply this method to two systems, the quantum spin Hall insulator and a staggered topological superconductor, and show that (spin) Chern numbers are accurately reproduced. The measurements required for constructing the component winding numbers also enable one to probe the entanglement spectrum with respect to component partitions. Our method is particularly suited to experiments with cold atoms in optical lattices where time-of-flight images can give direct access to the relevant observables. © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft., This work was supported by EPSRC, Spanish MINECO Project FIS2012–33022, Beca FPU no. AP 2009–1761, and CAM research consortium QUITEMAD S2009-ESP-1594.

We present tight-binding models of 3D topological superconductors in class DIII that support a variety of winding numbers. We show that gapless Majorana surface states emerge at their boundary in agreement with the bulk-boundary correspondence. At the presence of a Zeeman field, the surface states become gapped and the boundary behaves as a 2D superconductor in class D. Importantly, the 2D and 3D winding numbers are in agreement, signifying that the topological phase of the boundary is induced by the phase of the 3D bulk. Hence, the boundary of a 3D topological superconductor in class DIII can be used for the robust realization of localized Majorana zero modes.

Topological quantum computation started as a niche area of research aimed at employing particles with exotic statistics, called anyons, for performing quantum computation. Soon it evolved to include a wide variety of disciplines. Advances in the understanding of anyon properties inspired new quantum algorithms and helped in the characterisation of topological phases of matter and their experimental realisation. The conceptual appeal of topological systems as well as their promise for building fault-tolerant quantum technologies fuelled the fascination in this field. This `focus on' brings together several of the latest developments in the field and facilitates the synergy between different approaches., Comment: 5 pages, Editorial for NJP Focus Issue on Topological Quantum Computation, some references added, minor modifications

Within the framework of a local expansion of the logarithm of the O (N)σ-model vacuum functional, valid for slowly varying fields, the modified Virasoro algebra is studied. The operatorlike central charge term is given, up to second order, for a general functional.

For slowly varying fields the vacuum functional of a quantum field theory may be expanded in terms of local functionals. This expansion satisfies its own form of the Schr\"odinger equation from which the expansion coefficents can be found. For scalar field theory in 1+1 dimensions we show that this approach correctly reproduces the short-distance properties as contained in the counter-terms. We also describe an approximate simplification that occurs for the Sine-Gordon and Sinh-Gordon vacuum functionals., Comment: 21 pages, LaTeX, 8 figures

The Coulomb branch of the potential between two static colored sources is calculated for the Yang-Mills theory using the electric Schroedinger representation., Comment: 4 pages, no figures, REVTEX, as in Phys. Lett. B

To use quantum systems for technological applications we first need to preserve their coherence for macroscopic timescales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. Here we review the state-of-the-art developments in this field in an informative and pedagogical way. We give the main principles of self-correcting quantum memories and we analyze several milestone examples from the literature of two-, three- and higher-dimensional quantum memories., Comment: v2,3,4 - Final author copy; to appear in Rev. Mod. Phys.; 55 pages, 29 figures, 254 references; improvements include new sections on the Curie-Weiss model, open questions on interacting anyon models, and a discussion on SPT phases. Criteria for self correction is simplified, and the discussion on the 4D toric code is extended. Other changes are made following new work since submission of v1

For slowly varying fields on the scale of the lightest mass the logarithm of the vacuum functional of a massive quantum field theory can be expanded in terms of local functionals satisfying a form of the Schrodinger equation, the principal ingredient of which is a regulated functional Laplacian. We extend a previous work to construct the next to leading order terms of the Laplacian for the Schrodinger equation that acts on such local functionals. Like the leading order the next order is completely determined by imposing rotational invariance in the internal space together with closure of the Poincare algebra., Comment: 7 pages, Latex, no figures

Abelian Chern-Simons-Maxwell theory can emerge from the bosonisation of the 2+1-dimensional Thirring model that describes interacting Dirac fermions. Here we show how the Thirring model manifests itself in the low energy limit of a two-dimensional tight-binding model of spinless fermions. To establish that we employ a modification of Haldane's model, where the "doubling" of fermions is rectified by adiabatic elimination. Subsequently, fermionic interactions are introduced that lead to the analytically tractable Thirring model. By local density measurements of the lattice fermions we can establish that for specific values of the couplings the model exhibits the confining 2+1-dimensional QED phase or a topological ordered phase that corresponds to the Chern-Simons theory. The implementation of the model as well as the measurement protocol are accessible with current technology of cold atoms in optical lattices., Comment: 4.5 pages, 2 figures, version to appear in PRL

For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional of a massive quantum field theory can be expanded in terms of local functionals satisfying a form of the Schrodinger equation, the principal ingredient of which is a regulated functional Laplacian. We construct to leading order a Laplacian for the O(N) sigma-model that acts on such local functionals. It is determined by imposing rotational invariance in the internal space together with closure of the Poincare algebra.

Here, we provide a simple Hubbard-like model of spin-$1/2$ fermions that gives rise to the SU(2) symmetric Thirring model that is equivalent, in the low-energy limit, to Yang-Mills-Chern-Simons model. First, we identify the regime that simulates the SU(2) Yang-Mills theory. Then, we suitably extend this model so that it gives rise to the SU(2) level $k$ Chern-Simons theory with $k\geq2$ that can support non-Abelian anyons. This is achieved by introducing multiple fermionic species and modifying the Thirring interactions, while preserving the SU(2) symmetry. Our proposal provides the means to theoretically and experimentally probe non-Abelian SU(2) level $k$ topological phases., Comment: 4.5 pages, 3 figures, published version

The Chern number ν, as a topological invariant that identifies the winding of the ground state in the particle-hole space, is a definitive theoretical signature that determines whether a given superconducting system can support Majorana zero modes. Here we show that such a winding can be faithfully identified for any superconducting system (p wave or s wave with spin-orbit coupling) through a set of time-of-flight measurements, making it a diagnostic tool also in actual cold-atom experiments. As an application, we customize the measurement scheme for a chiral topological model of spinless fermions. The proposed model only requires the experimentally accessible s-wave pairing and staggered tunneling that mimics spin-orbit coupling. By adiabatically connecting this model to Kitaev's honeycomb lattice model, we show that it gives rise to ν=±1 phases, where vortices bind Majorana fermions, and ν=±2 phases that emerge as the unique collective state of such vortices. Hence, the preparation of these phases and the detection of their Chern numbers provide an unambiguous signature for the presence of Majorana modes. Finally, we demonstrate that our detection procedure is resilient against most inaccuracies in experimental control parameters as well as finite temperature. © 2013 American Physical Society., This work was supported by EP-SRC and by Spanish MICINN Project FIS2009-10061, Beca FPU No. AP 2009-1761, CAM research consortium QUITEMAD S2009-ESP-1594

In this work we present an optical lattice setup to realize a full Dirac Hamiltonian in 2+1 dimensions. We show how all possible external potentials coupled to the Dirac field can arise from perturbations of the existing couplings of the honeycomb lattice pattern. This greatly simplifies the proposed implementations, requiring only spatial modulations of the intensity of the laser beams to induce complex non-Abelian potentials. We finally suggest several experiments to observe the properties of the quantum field theory in the setup. © 2012 Elsevier Inc., This work has been funded by Spanish MICINN Project FIS2009-10061, FPU grant No.AP 2009-1761, CAM research consortium QUITEMAD S2009- ESP-1594, a Marie Curie Intra European Fellowship, CSIC Grant JAE-INT- 1072, the POLATOM network and the Royal Society.

Here we investigate the effect lattice geometry has on the lifetime of two-dimensional topological quantum memories. Initially, we introduce various lattice patterns and show how the error-tolerance against bit-flips and phase-flips depends on the structure of the underlying lattice. Subsequently, we investigate the dependence of the lifetime of the quantum memory on the structure of the underlying lattice when it is subject to a finite temperature. Importantly, we provide a simple effective formula for the lifetime of the memory in terms of the average degree of the lattice. Finally, we propose optimal geometries for the Josephson junction implementation of topological quantum memories., 12 pages, 6 figures